Term Rewriting System R:
[x, y]
f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(f(x)) -> F(g(f(x), x))
F(f(x)) -> G(f(x), x)
F(f(x)) -> F(h(f(x), f(x)))
F(f(x)) -> H(f(x), f(x))
H(x, x) -> G(x, 0)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

F(f(x)) -> F(h(f(x), f(x)))
F(f(x)) -> F(g(f(x), x))


Rules:


f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(f(x)) -> F(g(f(x), x))


The following usable rules for innermost can be oriented:

g(x, y) -> y
f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
h(x, x) -> g(x, 0)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(F(x1))=  x1  
  POL(f(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
f(x1) -> f(x1)
g(x1, x2) -> x2
h(x1, x2) -> x1


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Argument Filtering and Ordering


Dependency Pair:

F(f(x)) -> F(h(f(x), f(x)))


Rules:


f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(f(x)) -> F(h(f(x), f(x)))


The following usable rules for innermost can be oriented:

h(x, x) -> g(x, 0)
f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(h)=  0  
  POL(F(x1))=  x1  
  POL(f(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
f(x1) -> f(x1)
h(x1, x2) -> h
g(x1, x2) -> x2


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳AFS
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rules:


f(f(x)) -> f(g(f(x), x))
f(f(x)) -> f(h(f(x), f(x)))
g(x, y) -> y
h(x, x) -> g(x, 0)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes